Batch computing and stream computing are two important forms of big data computing. The research and discussions on batch computing in big data environment are comparatively sufficient. But how to efficiently deal with stream computing to meet many requirements, such as low latency, high throughput and continuously reliable running, and how to build efficient stream big data computing systems, are great challenges in the big data computing research. This paper provides a research of the data computing architecture and the key issues in stream computing in big data environments. Firstly, the research gives a brief summary of three application scenarios of stream computing in business intelligence, marketing and public service. It also shows distinctive features of the stream computing in big data environment, such as real time, volatility, burstiness, irregularity and infinity. A well-designed stream computing system always optimizes in system structure, data transmission, application interfaces, high-availability, and so on. Subsequently, the research offers detailed analyses and comparisons of five typical and open-source stream computing systems in big data environment. Finally, the research specifically addresses some new challenges of the stream big data systems, such as scalability, fault tolerance, consistency, load balancing and throughput.